Calculating the Depth of a Submarine's Dive: A Trigonometric Approach

Welcome to our guide on determining the depth a submarine reaches as it dives at a specific angle from the water surface. This article will walk you through a step-by-step process using trigonometry to find the depth of the submarine after it has traveled a certain distance. We will also explore an alternative method to validate our results.

The Problem at Hand

A submarine dives from the water surface at an angle of 31 degrees below the horizontal, traveling a straight path of 75 meters long. The challenge is to determine how far the submarine is below the water surface once it has completed this portion of its journey.

Step-by-Step Guide Using Trigonometry

To solve this problem, we will utilize fundamental principles of trigonometry. Here’s the step-by-step approach:

Identifying the Components of the Dive

The distance traveled (hypotenuse, H) is 75 meters. The angle of descent (θ) is 31 degrees.

Calculating the Vertical Component Depth

The vertical depth d can be calculated using the sine function:

d H × sinθ

Where:

H is the hypotenuse, which is 75 meters. θ is the angle below the horizontal, which is 31 degrees.

Substituting and Calculating the Depth

Let's break this down step by step:

First, substitute the values into the equation: d 75 m × sin 31° Next, calculate the sine of 31 degrees: sin 31° ≈ 0.5150 (using a calculator) Finally, calculate the depth: d 75 m × 0.5150 ≈ 38.625 m

Conclusion

The submarine is approximately 38.63 meters below the water surface.

Alternative Method Using Cosine

Even though our primary calculation used the sine function, we can also use the cosine function to validate our results. The cosine function calculates the horizontal displacement of the submarine.

Using Cosine to Calculate Horizontal Displacement

The horizontal displacement can be calculated as:

base H × cosθ

Substituting the values:

base 75 m × cos 31° Using a calculator, cos 31° ≈ 0.8572 base 75 m × 0.8572 ≈ 64.287 m

Thus, the submarine is approximately 64.287 meters horizontally from the initial point.

Vertical Depth Using Sine

Using the sine function to find the vertical depth again:

Calculate the vertical depth using the same sine function: Altitude 75 m × sin 31° ≈ 38.627 m

This confirms our earlier result for the vertical depth.

Visualization and Final Thoughts

Let’s visualize the problem as a right triangle with the horizontal base on the water surface, the vertical side representing the depth, and the hypotenuse being the path traveled by the submarine.

In summary, the submarine is approximately 38.63 meters below the water surface after traveling 75 meters at an angle of 31 degrees below the horizontal.

If you have any questions or need more detailed explanations, feel free to reach out. Understanding these principles can help in various marine and industrial applications.